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\usepackage{amsmath, amssymb} % 数学公式与符号
\usepackage{graphicx, color, url}
\usepackage{enumitem}

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\setlength{\droptitle}{-2cm} % 标题上移

\title{《基础复分析》第1章复数 - 部分习题解答}
\author{CGZ ET AL}

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\begin{document}
\maketitle 

%% 《基础复分析》习题一

\begin{enumerate}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %1

% 计算下列各数:
%  $\sqrt{i}$
%  $\sqrt{-i}$
%  $\sqrt{1+i}$
%  $\sqrt[4]{-1}$
%  $\sqrt[6]{i}$
%  $\sqrt[6]{-i}$
%  $\sqrt{\frac{1-i\sqrt{3}}{2}}$
    

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %2

% 解二次方程 $x^2 + (\alpha + i\beta)x + \gamma + i\delta = 0$.


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %3

% 计算 $$\frac{z}{z^2+1}$$ 其中 $z=x+iy$ 或者 $z=x-iy$, 并验证两个结果共轭。


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %4

% 求下列复数的模:
%  $$-2i(3+i)(2+4i)(1+i)$$
%  $$\frac{(3+4i)(-1+2i)}{(-1-i)(3-i)}$$


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\item[5.]  %5

当 $|a|=1$ 或者 $|b|=1$, 且 $\bar{a}b \neq 1$ 时, 证明
$$
\left|\frac{a-b}{1-\bar{a}b}\right| = 1.
$$

\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}


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\item[6.]  %6

当 $|a|<1$ 且 $|b|<1$ 时, 证明
$$
\left|\frac{a-b}{1-\bar{a}b}\right| < 1.
$$
    
\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}



% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %7
% 当 $|a_i|<1$, $\lambda_i \geq 0$ ($i=1,2,\cdots,n$) 且 $\lambda_1+\lambda_2+\cdots+\lambda_n=1$ 时, 证明
%     $$
%     |\lambda_1 a_1 + \lambda_2 a_2 + \cdots + \lambda_n a_n| < 1.
%     $$


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %8
    
% 证明存在复数 $z$ 满足 $|z-a|+|z+a|=2|c|$ 当且仅当 $|a| \leq |c|$. 

% 如果条件成立, 求 $|z|$ 的最大值和最小值。


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\item[9.]  %9
    
求点 $a \in \mathbb{C}$ 关于坐标轴分角线的对称点。

\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}


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\item[10.]  %10
    
证明点 $a_1, a_2, a_3$ 为等边三角形的三个顶点当且仅当 

$$
a_1^2 + a_2^2 + a_3^2 = a_1a_2 + a_2a_3 + a_3a_1.
$$

\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item[11.]  %11
    
% 设 $a$ 和 $b$ 为正方形的两个顶点，求另外两个顶点。


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% \item  %12
    
% 设一个三角形的三个顶点分别为 $a_1, a_2, a_3$, 求外接圆的圆心和半径。


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\item[13.]  %13
    
证明三角形的内角和为 $\pi$.

\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %14
    
% 将 $\cos 3\theta$, $\cos 4\theta$ 与 $\sin 5\theta$ 用 $\cos \theta$ 和 $\sin \theta$ 表示。


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %15
    
% 简化 

% $$
% 1 + \cos \theta + \cos 2\theta + \cdots + \cos n\theta
% $$ 

% 和 

% $$
% 1 + \sin \theta + \sin 2\theta + \cdots + \sin n\theta
% $$

% 其中 $n \geq 2$ 为正整数。


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\item[16.]  %16
    
用代数形式表示 5 次单位根和 10 次单位根。

\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %17
    
% 设 

% $$
% \omega = \cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n}
% $$

% 其中 $n \geq 2$ 为正整数。

% (1) 证明对任意不是 $n$ 的整数倍的整数 $k$, 有 

% $$
% 1 + \omega^k + \cdots + \omega^{(n-1)k} = 0
% $$

% (2) 求值

% $$
% 1 - \omega^k + \omega^{2k} - \cdots + (-1)^{n-1} \omega^{(n-1)k}$$



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\item[18.]  %18
    
证明平行四边形的对角线相互平分而菱形的对角线相互正交。

\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %19
    
% 证明一个圆的平行弦的中点在垂直于这些弦的直径上。


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %20

% 证明过 $a \in \mathbb{C}$ 与 $\frac{1}{\bar{a}}$ 的所有圆周都与单位圆周正交。


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\item[21.]  %21
    
分析方程 $az + b\bar{z} + c = 0$ 所表示的几何图形。

\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}


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\item[22.]  %22
    
证明点 $z$ 和 $z'$ 位于 Riemann 球面上一条直径的两个端点当且仅当 $zz' = -1$.

\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}


% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %23
    
% 一个立方体的所有顶点都在单位球面上，各棱平行于坐标轴，求各顶点的球极投影的像。


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\item[24.]  %24
    
设 $z, z' \in \mathbb{C}$ 为不同的点，$X, X'$ 为它们在球极投影下的原像，$N = (0, 0, 1)$. 证明 $\triangle NXX'$ 与 $\triangle NZZ'$ 相似。

\vspace{0.2cm}

{\color{red}解答：
\begin{enumerate}[label=(\arabic*)]
\item  
\item  
\item  
\end{enumerate}

}

\vspace{0.2cm}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% \item  %25
    
% 求圆心为 $a$、半径为 $R$ 的圆周在球极投影下的原像的半径。



\end{enumerate}

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\end{document}

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